Is the localization transition observed for interacting, strongly disordered bosons in two-dimensional optical lattices a genuine dynamical phase transition between a thermalizing and a many-body localized (MBL) phase, or a sharp finite-time and finite-size crossover? This question, posed by the first site-resolved two-dimensional MBL experiment (Choi et al., Science 2016), remains unresolved. Its full resolution requires establishing whether a non-ergodic localized phase survives the thermodynamic and infinite-time limit in two dimensions despite avalanche instabilities, and, if it does, determining the transition's critical properties — dynamical critical exponent, scaling laws, and Griffiths subdiffusive regime — together with a direct eigenstate-level characterization and a quantification of how weak environmental coupling limits the observable lifetime of localization. A decade of theory, numerics, and experiments has produced conflicting evidence in both directions, and no component of this characterization is settled.

Background

Statistical mechanics rests on the expectation that an isolated, generic quantum many-body system serves as its own heat bath: after long evolution, local observables reach thermal values, as formalized by the eigenstate thermalization hypothesis (ETH), which states that individual high-energy eigenstates already encode thermal equilibrium. Strong quenched disorder can break this paradigm. Building on Anderson localization of single particles, Basko, Aleiner and Altshuler (2006) argued that interactions alone do not restore transport at strong disorder: interacting disordered systems can enter a many-body localized (MBL) phase, in which transport is blocked, local memory of an initial state persists forever, and eigenstates violate ETH. In one dimension this picture is supported by extensive numerics, by a phenomenology of local integrals of motion (LIOMs) — quasi-local conserved operators that label the localized phase, with dephasing between them producing a characteristic logarithmic-in-time growth of entanglement entropy — and by ultracold-atom experiments that measure the 'imbalance', i.e. the persistence of an initially prepared density pattern, and even entanglement-related correlations (Lukin et al., 2019). In one dimension, the MBL transition is described by scaling theories in which a diverging length scale is accompanied by a diverging timescale; the associated dynamical critical exponent, governing the power-law divergence of relaxation times as the transition is approached, and subdiffusive Griffiths regimes — slow transport caused by rare, locally weakly disordered regions — are central elements of the one-dimensional theory (Abanin, Altman, Bloch, Serbyn, 2019). Dimensionality is decisive: for noninteracting particles, all states are localized in two dimensions for arbitrarily weak disorder, but interactions dramatically change the problem, and analytical and exact numerical tools are essentially confined to one dimension because the Hilbert space grows exponentially with system size. The experiment of Choi et al. (2016) realized a disordered two-dimensional Bose–Hubbard model (interacting bosons hopping in an optical lattice with a two-dimensional random onsite disorder potential, generated by projecting computer-generated random patterns with a digital mirror device, with a short correlation length set by the imaging numerical aperture) at scales far beyond classical simulation, and observed, through site-resolved imbalance measurements at evolution times of roughly 190τ190\tau (with τ\tau the tunneling time), a transition from a thermalizing to a localized regime, with evidence for a diverging length scale and a critical disorder strength that shifts when the interaction energy is reduced. However, the authors emphasized the open issues: an experiment at finite times cannot distinguish a true asymptotic phase transition from a sharp crossover; longer-time and larger-scale data would be needed to extract dynamical critical behavior; eigenstate properties (entanglement growth, LIOMs, site-resolved density correlations, Griffiths effects) were not probed; noise-induced heating, coupling to higher bands, and atom loss bound the accessible times and could eventually relax the quasi-steady state; and higher-dimensional MBL theory was (and remains) incomplete. The problem acquired a sharper theoretical edge with the avalanche mechanism of De Roeck and Huveneers (2017): a rare thermal inclusion in a localized background can, in dimensions d>1d>1, resonantly absorb neighboring localized regions, thermalize them, and thereby grow without bound — an instability that would preclude any true MBL phase in two dimensions, leaving only long-lived transient localization. Subsequent numerical work (Potirniche, Banerjee, Altman, 2019) supported this scenario, while stabilization mechanisms have also been proposed (Foo et al., 2023, showed that Gaussian — super-exponential — localization of LIOMs, natural in the confining potentials of experiments, pushes the avalanche critical dimension from d=1d=1 to d=2d=2), and independent two-dimensional numerics found MBL signatures in strongly disordered two-dimensional models (Bose–Hubbard, Heisenberg, and quantum-Monte-Carlo-amenable constructions) (Wahl, Pal, Simon, 2019; Decker, Kennes, Karrasch, 2022; Swain et al., 2025). Two-dimensional experiments have since measured slow relaxation in quasiperiodic disorder (Bordia et al., 2017), bath-induced delocalization (Rubio-Abadal et al., 2019), and finite-size scaling of the imbalance on optical-lattice and superconducting-simulator platforms (Hur et al., 2025; Li et al., 2025), with random and quasiperiodic disorder showing opposite system-size trends. The question of whether the observed two-dimensional 'MBL' is a phase or a crossover, and what its critical properties are, is therefore a precise and actively contested open problem.

Problem Statement

For interacting quantum many-body systems with strong quenched disorder in two spatial dimensions — as realized by bosons in a disordered two-dimensional optical lattice (the two-dimensional disordered Bose–Hubbard model) — determine whether the observed transition between a thermalizing and a localized regime is a genuine dynamical phase transition, i.e. whether a many-body localized phase with persistent non-ergodic dynamics survives the thermodynamic and infinite-time limit, or whether it is a finite-time, finite-size crossover that eventually thermalizes, as predicted by avalanche-instability arguments based on rare thermal inclusions. This overarching question comprises the following jointly linked components, all originating in the source experiment's outlook: (1) establish whether the quasi-steady localized regime (observed at evolution times of order 190τ190\tau, with τ\tau the lattice tunneling time, and at fixed system sizes) remains stable as time and system size grow, including how the answer depends on the disorder statistics (random versus quasiperiodic) and on the interaction strength, and whether the apparent critical disorder drifts without bound; (2) if the transition exists, determine each of its critical properties — the diverging length and timescales, the dynamical critical exponent governing relaxation times near criticality, and the presence and extent of Griffiths-type rare-region effects and subdiffusive transport; (3) if the transition exists, characterize the localized phase at the eigenstate level in two dimensions (logarithmic entanglement-entropy growth, local integrals of motion, and density–density correlations beyond averaged profiles); and (4) quantify how weak coupling to the environment and experimental imperfections (such as heating, low-frequency noise, atom loss, coupling to higher bands) bound the localization lifetimes that any real two-dimensional system can exhibit, separating genuine asymptotic physics from observationally limited transients.

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